A Muscle-Reflex Model That Encodes Principles of Legged
Mechanics Produces Human Walking Dynamics and
Muscle Activities
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Citation
Geyer, Hartmut, and Hugh Herr. “A Muscle-Reflex Model That
Encodes Principles of Legged Mechanics Produces Human
Walking Dynamics and Muscle Activities.” IEEE Transactions on
Neural Systems and Rehabilitation Engineering 18.3 (2010):
263–273. Web. ©2010 IEEE.
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http://dx.doi.org/10.1109/TNSRE.2010.2047592
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Institute of Electrical and Electronics Engineers
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http://hdl.handle.net/1721.1/70926
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IEEE TRANSACTIONS ON NEURAL SYSTEMS AND REHABILITATION ENGINEERING, VOL. 18, NO. 3, JUNE 2010
263
A Muscle-Reflex Model That Encodes Principles
of Legged Mechanics Produces Human Walking
Dynamics and Muscle Activities
Hartmut Geyer and Hugh Herr, Member, IEEE
Abstract—While neuroscientists identify increasingly complex
neural circuits that control animal and human gait, biomechanists
find that locomotion requires little control if principles of legged
mechanics are heeded that shape and exploit the dynamics of
legged systems. Here, we show that muscle reflexes could be vital
to link these two observations. We develop a model of human
locomotion that is controlled by muscle reflexes which encode
principles of legged mechanics. Equipped with this reflex control,
we find this model to stabilize into a walking gait from its dynamic
interplay with the ground, reproduce human walking dynamics
and leg kinematics, tolerate ground disturbances, and adapt
to slopes without parameter interventions. In addition, we find
this model to predict some individual muscle activation patterns
known from walking experiments. The results suggest not only
that the interplay between mechanics and motor control is essential to human locomotion, but also that human motor output could
for some muscles be dominated by neural circuits that encode
principles of legged mechanics.
Index Terms—Balance, feedback control, legged locomotion.
I. INTRODUCTION
EGGED locomotion of animals and humans is controlled
by a complex network of neurons. Proposed in the early
20th century [1] and firmly established for animals today [2],
the central pattern generator (CPG) forms the basis of this network. In the current view, the CPG consists of layers of neuron
pools in the spinal cord [3] which, through other neuron pools
channeling muscle synergies, provide rhythmic activity to the
leg extensor and flexor muscles [4], [5] sufficient to generate
stepping movements, even in the absence of spinal reflexes [6].
Spinal reflexes are nevertheless part of this complex network,
contributing to the selection of locomotive patterns, the timing
of the extensor and flexor activities, and the modulation of the
CPG output [2], [7], [8]. Using this combination of a central pattern generation and modulating reflexes, neuromuscular models
of lampreys [9], salamanders [10], cats [11]–[13], and humans
L
Manuscript received February 20, 2009; revised August 25, 2009; accepted
February 02, 2010. First published April 08, 2010; current version published
June 09, 2010. The work of H. Geyer was supported by an EU Marie-Curie
Fellowship (MOIF-CT-20052-022244).
H. Geyer is with the Robotics Institute, Carnegie Mellon University, Pittsburgh, PA 15213 USA (e-mail: hgeyer@cs.cmu.edu).
H. Herr is with the Media Laboratory, Massachusetts Institute of Technology,
Cambridge, MA 02139 USA (e-mail: hherr@media.mit.edu).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TNSRE.2010.2047592
[14], [15] have developed into essential tools for studying different control strategies in animal and human locomotion. The
emphasis of these models has been to reproduce the architecture
of the CPG and underlying reflexes suggested by experiments
[7]. Little attention has however been paid to understanding
how such architectures might represent or encode principles of
legged mechanics.
Several principles of legged mechanics indicate that the
seemingly complex task of locomotion control can largely
be simplified. One such principle is mechanical self-stability.
Using conceptual models of walking [16], [17] and running
[18], [19] that capture the main features of legged mechanics,
researchers have shown that legged locomotion self-stabilizes
without control interventions if the mechanical components
are properly tuned [20]–[23]. Walking and running robots
have demonstrated the practical relevance and control benefits
derived from mechanical self-stability [20], [24]–[26]. Another
such principle is the reliance on compliant leg behavior. It has
been shown that, if the legs behave similar to springs in stance,
walking and running are only two out of many gaits of the
same mechanical system which naturally emerge at different
speeds without the need for a gait-specific control [27]. But
it remains unclear if and how these and other principles of
legged mechanics that simplify the control of locomotion are
integrated into human motor control.
The natural candidates for such an integration are spinal reflexes, because they can link sensory information about legged
mechanics directly into the activation of the leg muscles via
alpha motoneurons, bypassing central inputs. For instance, in
models of neuromuscular control, positive force feedback of leg
extensor muscles has been shown to not only play an important
role in load-bearing [28], but also generate compliant leg behavior in stance [29], suggesting that this key mechanical behavior can effectively be encoded in human motor control by
a single muscle reflex. These conceptual models are however
too simplistic to really compare predictions about motor control
with the activity patterns observed for individual leg muscles.
To clarify the influence of legged mechanics on human motor
control, we here develop a more detailed neuromuscular human
model that expands on the idea of encoding principles of legged
mechanics in autonomous muscle reflexes. The model represents the human body with a trunk and two three-segment legs.
Each leg is actuated by seven Hill-type muscles that permit a
direct comparison with prominent muscles of the human leg.
In Section II, we detail how this model and its control evolve
from the reliance on compliant leg behavior as a core principle
1534-4320/$26.00 © 2010 IEEE
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IEEE TRANSACTIONS ON NEURAL SYSTEMS AND REHABILITATION ENGINEERING, VOL. 18, NO. 3, JUNE 2010
Fig. 1. Model evolution. Stance leg: (a) Compliant leg behavior as key to walk and run is generated (b) by driving the SOL and the lumped vasti group muscles
(VAS) with positive force feedbacks F+. (c) To prevent knee overextension the biarticular GAS muscle is added using F+, and the VAS gets inhibited if the knee
extends beyond a 170 threshold. To prevent ankle overextension, the TA muscle is added whose pulling of the ankle joint into a flexed position by positive length
feedback L+ is suppressed under normal stance conditions by negative force feedback F- from soleus. Trunk: (d) The trunk is driven into a reference lean with
respect to the vertical by the HFL and co-activated hip extensor muscles (GLU, HAM) of the stance leg, where the biarticular HAM prevents knee overextension
resulting from hip extensor moments. The trunk reflexes are modulated by the load the stance leg bears. Swing leg: (e) The landing of the other (leading) leg
initiates swing by adding/subtracting a constant stimulation to HFL/GLU, respectively, and by suppressing VAS proportionally to the load borne by the other leg.
(f) The actual leg swing is facilitated by HFL using L+ until it gets suppressed by L- of HAM. HFL’s stimulation is biased dependent on the trunk’s lean at takeoff.
Moreover, using F+ for GLU and HAM retracts and straightens the leg toward the end of swing. Finally, the now unsuppressed L+ of TA drives the ankle to a
flexed position.
of legged locomotion [18], [19], [27]. Throughout this process,
we encode in muscle reflexes more principles of legged mechanics, for instance, to avoid joint overextension of segmented
legs [30], [31], or to improve gait stability [23], [32]–[34].
Comparing the model’s behavior with kinetic, kinematic, and
electromyographic evidence from the literature, we show in
Section III that a neuromuscular model equipped with this
principle-based motor control not only can produce biological
walking mechanics, but also predicts the observed activation
patterns of some individual muscles. We further show that this
reflex control allows the model to tolerate ground disturbances
and to adapt to slopes without parameter interventions. Finally,
we discuss in Section IV the implications of our results.
II. HUMAN MODEL
The conceptual basis for the human model is the bipedal
spring-mass model [Fig. 1(a)], which simplifies human locomotion to a point mass that travels on two massless spring
legs. Despite its simplicity, the bipedal spring-mass model
reproduces the center-of-mass dynamics observed in human
walking and running, unifying both gaits in one mechanical
framework based on compliant leg behavior in stance [27].
To translate this conceptual model into a neuromuscular one,
which better reflects human morphology, three main steps are
required. First, the springs must be replaced with segmented
legs, and compliant stance behavior must be generated by
extensor muscles spanning the ankle and knee. Second, the
point mass must be replaced with a trunk, and hip muscles must
be added for its balance control. And third, swing leg control
must be added to enable this model to enter cyclic locomotion.
In this section, we detail how the structure and control of the
human model is guided by these three main steps. A major part
of this model evolution is driven by principles of legged mechanics that we encode in muscle reflexes. Throughout this section, we try to motivate these reflexes with neurophysiological
evidence from the literature.
A. Replacing the Leg Springs With Segmented Legs
In an earlier study, it was shown that positive force feedback
(F+) of the extensor muscles, a spinal reflex during stance
observed in cats [35] and suggested in humans [28], [36], can
effectively generate compliant behavior in neuromuscular legs
[29]. We thus replace each spring of the bipedal spring-mass
model with a segmented leg that has thigh, shank and foot
(Table IV), and add a soleus muscle (SOL) and a vasti muscle
group (VAS) (Table II), both generating their own muscle
activity in stance using F+ [Fig. 1(b)]. We model this force
reflex in the same way as in [29]. With F+, the stimulation
of a muscle is the sum of a prestimulation
, and
and gained
force
:
the muscle’s time-delayed
. Details on how reflex
parameters were chosen are provided in the result section and
Appendix I. Appendix II describes how muscle stimulation
translates into muscle force, and Appendix III explains the
model’s musculoskeletal connections, joint architecture, and
mass distribution.
GEYER AND HERR: A MUSCLE-REFLEX MODEL THAT ENCODES PRINCIPLES OF LEGGED MECHANICS
Although the segmentation into thigh, shank and foot is essential to represent the structure of the human leg, it also introduces a control problem during leg compression if the joints
are compliant [30], [31], as guaranteed by F+ of SOL and VAS.
In segmented legs, the knee and ankle torques, and , obey
, where
and
are
the static equilibrium
the perpendicular distances from the knee and the ankle to the
, respecvector of the leg ground reaction force (GRF),
tively. In effect, a large extension torque at one joint forces the
, threatening its overextension (for deother joint closer to
tails see [30]).
We counter this tendency to overextend at the knee or the
ankle by adding the gastrocnemius (GAS) and tibialis anterior
(TA) muscles [Fig. 1(c)]. Like SOL and VAS, the biarticular
GAS uses F+ during the stance period of gait. This muscle reflex not only prevents knee hyperextension resulting from large
extension torques at the ankle, but also contributes to generating
an overall compliant leg behavior. In contrast, the monoarticular TA uses local positive length feedback (L+) with
where
is the
is a length offset. Flexing the foot,
TA fiber length and
TA’s stretch reflex L+ prevents the ankle from overextending
when large knee torques develop. However, this reflex is not required if sufficiently active ankle extensors preserve the torque
equilibrium between the knee and ankle. To avoid that the TA
unnecessarily fights the SOL in this situation, we inhibit the TA
stimulation with a negative force feedback (F-) from the SOL,
resulting in
.
The implemented TA control is supported by evidence from
reflex experiments. These experiments show that a large TA
stretch response is present in swing, but suppressed mainly
when TA is silent in stance [37], and it has been suggested that
disynaptic Ia reciprocal pathways from ankle plantar flexors to
dorsiflexors are responsible for this inhibition [38].
Without direct support from neurophysiological experiments,
we further protect the knee from hyperextension by inhibiting
VAS if the knee extends beyond a 170 threshold,
, where
is a proportional gain,
, and
is the
and
knee angle. This reflex inhibition is only active if
the knee is actually extending. In humans, it would require the
sensory information from pressure cells around the knee joint
capsule to translate into knee position and velocity.
B. Replacing the Point Mass With a Trunk
For the next model evolution, we replace the point mass of
the bipedal spring-mass model with a trunk segment [Table IV,
Fig. 1(d)] that must be balanced during locomotion. Balancing
the trunk is generally regarded as a multisensor integration task
that mixes sensory information from the vestibular organs, visual cues, and proprioception from the leg muscles [39]. While
this complex integration seems critical to control standing, it
may not be required during locomotion [40]. In line with this
observation, [41] could stabilize the trunk of a human model
in walking only by activating the hip muscles proportional to
the velocity of the trunk and to its forward lean in the inertial
system.
265
We balance the trunk in a similar way. We add to each leg
a gluteus muscle group (GLU) and a HFL muscle group. The
GLU and the HFL are stimulated with a proportional-derivative signal of the trunk’s forward lean angle with respect to
, where
and
gravity,
are the proportional and derivative gains, and
is a reference
lean angle. This proportional-derivative trunk control can be interpreted as a reflex control that uses sensory information from
the vestibular organs; however, it is not based on a particular
principle of legged mechanics. In addition, we include the biarto
ticular hamstring muscle group (HAM) with
counter knee hyperextension that results from a large hip torque
developed by the GLU when pulling back the heavy trunk. Since
hip torques can only balance the trunk if the legs bear sufficient
weight on the ground, we modulate the stimulations of the GLU,
HAM, and HFL for each leg proportionally to the amount of
body weight it bears [shown as projection from the ipsilateral
thigh in Fig. 1(d)]. As a result, each leg’s hip muscles contribute
to the trunk’s balance control only during stance.
C. Adding Swing Leg Control
The human model’s muscle-reflex control so far generates
compliant leg behavior in stance while preventing joint overextension and balancing the trunk. To enable this model to enter
cyclic locomotion, we add swing leg control.
We assume that the functional importance of each leg in
stance reduces in proportion to the amount of body weight (bw)
borne by the other leg, and thus initiate swing already in double
support [Fig. 1(e)]. The human model detects which leg enters
stance last (contralateral leg), and inhibits F+ of the ipsilateral
leg’s VAS in proportion to the weight the contralateral leg bears,
where
is the weight gain and
the contralateral leg force. The contralateral inhibition allows
the knee to break its functional spring behavior and flex while
the ankle extends, pushing the leg off the ground and forward.
Thus the ankle push-off commonly regarded as a major principle of gait to smooth the transition from the double support
to the swing phase [16], [42] becomes an indirect outcome of
the inhibition at the knee implemented to eliminate compliant
leg behavior when it looses functional significance. In addition
to the indirect push-off, the model further initiates swing by
increasing the stimulation of the HFL, and decreasing that of
in double support.
the GLU, by a fixed amount
The implemented swing initiation reflects the current view
on the peripheral control of the stance-to-swing transition. This
view favors a mixed sensory input related to leg-unloading and
hip positioning [43], where the first input is always required
whereas the second one is more variable [44], and therefore, its
actual implementation is less clear. It could moreover be shown
that while unloading is essential, a direct input from the ipsilateral leg extensors via group-I afferents is not involved in the
stance-to-swing transition [45]. By contrast, recent experiments
on cockroach walking support a major role of the onset of another leg’s stance in triggering the first leg’s unloading [46].
During actual swing, we mainly rely on a leg’s ballistic motion [16]. The distal leg muscles SOL, GAS, and VAS are silent
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IEEE TRANSACTIONS ON NEURAL SYSTEMS AND REHABILITATION ENGINEERING, VOL. 18, NO. 3, JUNE 2010
in that phase. Only TA’s L+, introduced in Section II-A, remains
active to provide foot clearance with the ground.
We modulate the ballistic motion in two necessary ways
[Fig. 1(f)]. First, as the natural frequency of the purely ballistic
leg swing is too low to ensure a timely foot placement [16], the
model’s proximal HFL gets stimulated by its own stretch reflex
L+, facilitating leg protraction during swing. Such a homonymous reflex-shaping of hip flexor activities has been suggested
from experiments with decerebrate cats [47]; however, since
the required protraction speed depends on the trunk’s forward
lean, in the human model HFL’s L+ is biased by the trunk’s
at take off (TO), resulting in
pitch
.
Second, we improve gait stability by enforcing swing-leg
retraction. If legs reach and maintain a proper orientation
during swing, legged systems self-stabilize mechanically
into a gait cycle [20]–[22], [27]. The tolerance of this
mechanical self-stability against disturbances largely improves if a leg retracts before landing [23]. The human
model realizes this halt-and-retract strategy with three
muscle reflexes. One reflex inhibits the HFL’s L+ proportional to the stretch which the HAM receives in swing,
.
This negative length feedback L- compensates for the hip
rotation that results from the transfer of angular momentum
when the passive knee rotates into full extension during
leg protraction. The other two reflexes, F+ of the GLU,
, and F+ of
,
the HAM,
ensure that, dependent on the actual protraction momentum,
the swing leg not only halts, but also transfers part of this
momentum into leg lowering and retraction.
Some neurophysiological evidence exists to support the implemented reflex control for leg retraction, though mainly for the
hamstring. The excitation of the hamstring has been observed as
a recovery strategy in late swing lowering the leg and shortening
the step [48]. Moreover, this muscle group’s tendon jerk reflex
is enhanced in that phase, signaling a clear reflex contribution
to its activation [49]. A similar reflex activity for the glutei has
not been documented. Nor is a Ia reciprocal inhibition known
that projects from the hamstring to the hip flexors in swing.
Although the human model has no central pattern generator
(CPG) that feed-forwardly activates its muscles, it switches for
each leg between the different reflexes for stance and swing
using sensors at the ball and heel of each foot that detect ground
(Fig. 2). These sensors mimic mechanoreceptors in the foot,
which are suggested to be important for the control of phase
transitions in humans [43].
III. RESULTS
Because of the switches between stance and swing reflexes
based on ground detection, the model’s dynamic interaction
with its mechanical environment becomes a vital part of generating muscle activities. To clarify the influence of legged
mechanics on human motor control, we first try to make this
model walk like a human and then compare its predicted motor
output with muscle activations from the literature.
Fig. 2. Pattern generation. Instead of a central pattern, reflexes generate the
muscle stimulations S . Left (L) and right (R) leg have separate stance and
swing reflexes, which are selected based on contact sensing. The reflex outputs
depend on mechanical inputs M intertwining mechanics and motor control.
For the first part, we require the model (i) to produce cyclic
motions with GRFs similar to those of human walking, (ii)
to observe gait determinants relevant to sagittal-plane motion,
including early stance knee flexion, controlled plantar flexion,
powered plantar flexion and anterior–posterior flexion of the
trunk [42], and (iii) to demonstrate some robustness against
ground disturbances. We implement the model in Matlab
SimMechanics (ver. 2.7) and repeatedly start it from a typical
walking speed of 1.3 ms , manually tuning the reflex parameters to match our mechanical requirements. As initial values
for the reflex parameters, we use informed estimates. Note that
all results are presented for the final values, which we obtain
by maximizing (iii) constrained by (i) and (ii) (see Appendix I
for details on initial estimates and final values).
A. Walking Gait
Figs. 3 and 4 show the result of this manual reflex tuning. In
Fig. 3, the model starts with its left leg in stance and its right
leg in swing. Since the modeled muscle reflexes include signal
transport delays of up to 20 ms, all muscles are silent at first.
Because of these disturbed initial conditions, the model slightly
collapses and slows down in its first step [Fig. 3(a)]. It recovers
however in the next few steps, and walking self-organizes from
the dynamic interplay between model and ground. Here, the
vertical GRF of the legs in stance shows the M-shape pattern
characteristic for walking gaits [Fig. 3(b)], indicating similar
center-of-mass dynamics of model and humans for steady-state
walking. Here we consider the model to be in steady state only
if its joint kinematics vary 1e-4 degrees from stride to stride.
B. Steady-State Patterns of Joint Angles and Torques
The reflex model produces angle and torque trajectories that
are similar to those of human walking (Fig. 4). To quantify the
agreement, we use the maximum cross-correlation coefficients
of model and human trajectories (human data digitized from
[50] and then interpolated to 150 data points evenly distributed
from 0% to 100% stride), and the corresponding time shifts in
percent of stride if significantly different from zero (95% confishows a perfect agreement, whereas
dence interval) [51].
indicates no agreement. Because the model distinguishes
between stance and swing control, we split the comparison into
GEYER AND HERR: A MUSCLE-REFLEX MODEL THAT ENCODES PRINCIPLES OF LEGGED MECHANICS
267
Fig. 3. Walking self-organized from dynamic interplay with ground. (a) Snapshots of human model taken every 250 ms. Leg muscles shown only for the
: ankle, knee, and hip
right leg with dark color for activations 10%.
, 175 , 175 for left leg and 90 , 175 ,
angle (initial conditions:
140 for right leg). (b) Corresponding model GRFs normalized to body weight
(bw). Right and left leg GRFs shown in black and gray (30 Hz low-pass filtered),
with thick and thin traces marking the vertical and fore-aft components.
'
= 85
>
'
these two phases. The joint kinematics show a strong agreement for all joints in stance (
,
,
), and for the hip and knee in swing
.
. The
The ankle kinematics fit less well in that phase
difference is due mainly to maximizing the model robustness
against ground disturbances (compare Section III-D), which requires a rapid foot clearance not found in level walking. The
, but
joint torques nearly match for the ankle
and lesser still
show less agreement for the knee
for the hip
. The major difference
in the knee and hip torques occurs in early stance where, in the
model, knee extension torque is diminished, and hip extension
torque exaggerated and its onset delayed by about 5%. (Swing
torques are not compared; [50] only reports stance torques.)
C. Predicted Motor Output
Fig. 4 furthermore shows that the reflex model can not only
produce human walking dynamics and kinematics, but also predicts known activation patterns. In stance, the correlations between predicted and measured activation patterns lie within the
range observed in experiments [52] for all muscles. The patterns
,
) and GAS (
,
)
of SOL (
show the strongest agreement. The shift by 9% of stride in the
predicted patterns is caused by the continued activity of the
model’s plantar flexors until the end of stance, and is related to
the toe segment and associated muscles absent from the model.
In humans, ankle plantar flexion in late stance is supported by
toe flexors (and other small muscles crossing the subtalar joint)
[50], which lessens the load on triceps surae. The patterns of
,
) and HAM (
,
)
GLU (
share similar features and have similar -values and delays of
,
onset in the model. The predicted VAS pattern (
) captures the early stance activity in humans, but starts
from a lower initial activity and shows a second peak not seen
,
)
in experiments. The predicted TA pattern (
shares the lower initial activity, yet matches the remainder of
Fig. 4. Steady-state walking at 1.3 ms . Normalized to one stride from heelstrike to heel-strike of the same leg, the model’s steady-state patterns of muscle
activations, torques, and angles of (a) hip, (b) knee, and (c) ankle are compared
to human walking data (adapted from [50]). Vertical dotted lines around 60%
of stride indicate toe off. Abbreviations are given in Fig. 1. Compared muscles:
(i) adductor longus, (ii) upper gluteus maximum, (iii) semimembranosis, and
(iv) vastus lateralis.
TA’s pattern in experiments. Finally, the muscle activity of HFL
in stance.
shows the weakest agreement
In swing, the correlation reveals some experimental activation features unidentified by the model. The strongest agreement
, although its overall acis observed for the HAM
tivity is clearly too low in the model. The difference indicates
that the HAM force is overestimated during swing in the model,
which is supported by [53] who report that only the semitendinosus muscle of the hamstrings influences swing leg motion.
and TA
show
The patterns of HFL
similar levels of agreement. One clear difference in the TA patterns occurs in late swing, where activity stays about constant
in the model but rises in humans, preparing for stance [50]. The
and VAS
same feature is also lacking for GLU
in the model, showing a clear mismatch in motor
output.
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IEEE TRANSACTIONS ON NEURAL SYSTEMS AND REHABILITATION ENGINEERING, VOL. 18, NO. 3, JUNE 2010
Fig. 5. Slope adaptation. Approaching from steady-state walking at 1.3 ms , 14 strides of the human model are shown adapting to slope ascent and descent with
4 cm vertical steps. One stride is defined from heel-strike to heel-strike of the right leg. Shown are (a) snapshots of the model at heel-strike and toe-off of the right
leg, (b) right leg muscle activation patterns, and (c) GRFs (right and left leg in black and gray) normalized to bw.
The lack of stance preparation in the model explains the
observable differences between model and human walking in
stance. It causes the low initial stance activities of VAS and
TA, which in turn results in an increased knee flexion
and insufficient controlled plantar flexion
. As a consequence, the model’s trunk experiences a large forward tilt from
the insufficiently damped impact when the forefoot hits the
ground, requiring the hip muscles GLU and HAM to generate
to maintain trunk balance
exaggerated extension moments
.
D. Adaptation to Slopes
Despite its limited reflex control, the model shows robustness
against small ground disturbances (
4 cm) and can adapt to
4%) without parameter interventions. Fig. 5 proslopes (
vides an example in which the model encounters up (strides 2–6)
and down slopes (strides 9–12) (see supplementary animation 1
for a trial in which the model encounters irregular terrain and
longer slopes). No single control is responsible for this adaptation, but the dynamic interplay between legged mechanics and
motor control. For instance, the compliance and rebound of the
stance leg depends on how much load the leg extensors SOL,
GAS, and VAS experience, which guarantees that the leg yields
sufficiently to allow forward progression when going up, but
brakes substantially when going down (panels B and C). For
another example, the motion of the swing leg is accelerated by
the mechanical impact of the opposite leg, the forward lean of
the trunk, and an increased ankle push-off. These combined features ensure that the swing leg protracts enough when going up
and substantially so when going down (panel A), where the dynamic pull that GLU and HAM experience ensures that excess
rotation of the leg is converted into rapid retraction and straightening (panel B).
Note however that for the maximum slopes of 4% the model
is sensitive to how the swing foot hits the ground. If the toe
hits a step frontally when going up, or it touches the ground in
mid-swing when going down, the model trips and can eventually fall. In general, we observe the model behavior to be very
robust for the stance leg, but more sensitive to external disturbances and internal reflex adjustments for the swing leg (compare Table I in Appendix I for the sensitivity of the reflex parameters). For instance, if the model starts from an initial running
speed of about 3 ms , it manages some steps that resemble
human running, but eventually falls because the swing leg fails
(see supplementary animation 2).
IV. DISCUSSION
Our results suggest that mechanics and motor control cannot
be viewed separately in human locomotion. We started from
the assumption that principles of legged mechanics play an
important role in locomotion and developed the conceptual
spring-mass model, which explains the basic dynamics of
human locomotion, into a neuromuscular one that resembles
human morphology. For this development, we needed to encode several principles of legged mechanics with actuators
and control, which turned into muscles and reflexes. Besides
the generation of compliant stance-leg behavior [18], [19],
[27], these principles included the stabilization of segmented
chains against joint overextension when compressing in stance
GEYER AND HERR: A MUSCLE-REFLEX MODEL THAT ENCODES PRINCIPLES OF LEGGED MECHANICS
TABLE I
REFLEX PARAMETERS AND THEIR TOLERANCE. GAINS
AND
ARE
AND THE BODY WEIGHT. OFFSETS
ARE
NORMALIZED TO
. PRESTIMULATIONS
ARE 0.01 (NOT
SHOWN IN FRACTIONS OF
AND
OF THE
SHOWN) EXCEPT FOR THE STANCE VALUES
VAS AND OF THE TRUNK BALANCE MUSCLES HAM, GLU AND HFL
F
G
`
S
S
`
k
S
[30], [31], the indirect generation of ankle push-off [16], [42]
by eliminating compliant leg behavior in proportion to its
loss of functional significance in double support, the reliance
mainly on ballistic motions for the lower leg in swing [16],
and the improvement of gait stability by swing-leg retraction
[23], [32]–[34]. While more principles of legged mechanics do
certainly exist, the ones we implemented were sufficient for
the human model to enter cyclic motions. Taken separately,
these principles cannot account for human leg dynamics and
kinematics at the level of detail we investigated; and there
was no guarantee that taken together they would. However,
we found after tuning the resulting muscle reflexes that, by
combining these principles, human walking dynamics and leg
kinematics emerge (Figs. 3 and 4), and the model tolerates
ground disturbances and adapts to slopes without parameter
interventions (Fig. 5). Moreover, we found that the model
predicts some individual muscle activation patterns observed
in walking experiments (Fig. 4). These results suggest that the
interplay between mechanics and motor control is not only
important, but could for some muscles dominate human motor
output in locomotion.
Our findings support the view that centrally generated patterns of muscle activity may have limited functional relevance
to normal locomotion. While it is generally accepted that CPGs
can form a central drive for motor activity [4], [6], [54], their
functional role in human locomotion is debated [43], [55].
On one side, it has been shown that locomotor-like activity of
leg muscles can be evoked by tonic stimulation of the human
spinal cord, favoring the existence and functional relevance of
CPGs in man [5]. On the other side, the debate is fueled by the
lack of direct experimental evidence of human CPGs, and by a
continuing awareness that mechanics and motor control should
be intertwined [7]. For instance, back in 1969, Lundberg [56]
already suggested that, out of rather simple central patterns,
spinal reflexes could shape the complex muscle activities seen
in real locomotion. Refining this idea, Taga [57] later proposed
that, because “centrally generated rhythms are entrained by
sensory signals which are induced by rhythmic movements
[,] motor output is an emergent
of the motor apparatus
property of the dynamic interaction between the neural system,
269
the musculo-skeletal system, and the environment.” In support
of his claim, Taga [57] presented a neuromuscular model of
human locomotion that combined a CPG with sensory feedback. He demonstrated how basic gait can emerge from the
global entrainment between the rhythmic activities of the neural
and musculo-skeletal systems.
What the actual ratio of central and reflex inputs is that generates the motor output remains unclear, however [12], [58],
[59]. For instance, for walking cats, it has been estimated that
only about 30% of the muscle activity observed in the weight
bearing leg extensors can be attributed to muscle reflexes [60],
[61]. In humans, the contribution of reflexes to the muscle activities in locomotion seems to be more prominent. Sinkjaer et al.
estimated from unloading experiments that reflexes contribute
about 50% to the soleus muscle activity during stance in walking
[62]. More recently, Grey et al. found that the soleus activity
changes proportionally to changes in the Achilles tendon force,
suggesting a direct relationship between positive force feedback
and activity for this muscle [36]. Whether such a large reflex
contribution is present for all leg muscles is unclear. Perhaps
the motor control of humans shows the same proximo-distal gradient as the one Daley et al. proposed for birds. They concluded
from bird running experiments that proximal leg muscles are
mainly controlled by central inputs while distal leg muscles are
governed by reflex inputs due to higher proprioceptive feedback
gains and a larger sensitivity to mechanical effects [63]. Having
no CPGs, our model shows that no central input is required to
generate walking motions and muscle activities, suggesting that
reflex inputs which continuously mediate between the nervous
system and its mechanical environment may even take precedence over central inputs in the control of normal human locomotion.
Experiments will be needed to probe this conclusion.
Here, the principled approach detailed in this paper offers
an advantage over the more common approach attempting to
reverse-engineer human motor output. In many cases, neuromuscular models of animal and human locomotion mimic as
many neural structures as suggested by physiological evidence,
including CPGs, pattern formation and reflex networks [3],
[11]–[15], [57]. Although these models can be optimized to
generate locomotion steps, their predictive power is limited.
The functional relevance of their individual control elements
cannot be separated clearly. Nor can they reveal essential
control structures that lie still undiscovered. The principled
approach, by contrast, discards at first all the suggested control
structures. Synthesizing motor control element by element, it
allows to relate individual motor output to underlying mechanical function, and to make testable predictions about control
elements that have not yet been described in experiments.
Several muscle reflexes of the human model are currently not
backed by physiological evidence (compare Section II). They
provide testable predictions about a motor control that encodes
principles of legged mechanics.
While it is too early to draw definite conclusions about the
neural consequences of our modeling results, the technical merit
of the identified muscle-reflex control we demonstrate in a companion paper on the control of a powered ankle prosthesis.
270
IEEE TRANSACTIONS ON NEURAL SYSTEMS AND REHABILITATION ENGINEERING, VOL. 18, NO. 3, JUNE 2010
APPENDIX I
REFLEX CONTROL PARAMETERS
Initial values for the reflex parameters were obtained from
our previous study on reflex behavior, and from approximating
the trunk as an inverted pendulum and the swing leg as a double
pendulum driven at the hip. In detail, the F+ of SOL, GAS, and
VAS [Section II-A, Fig. 1(b) and (c)] had initial reflex gains
and prestimulations of 1% reported to generate reof
bound behavior [29], and L+ of TA was adjusted to dorsiflex the
ankle to 5 deg in 100 ms. For the trunk balance [Section II-B,
and , and the prestimulation
,
Fig. 1(d)], the gains
were initially set to balance and critically damp the inverted
trunk pendulum with a natural frequency of 2 Hz and a forward
lean of 5 deg (typical values in human walking), assuming
actuators with a maximum force of 3000 N and a lever of
was then adjusted so that one body weight
10 cm. The gain
fully suppressed muscle activation. For the swing leg control
of HFL and GLU had a
[Section II-C, Fig. 1(e) and (f)],
start value sufficient to generate a step from stand still of the
model. The L+ reflex gain of HFL was adjusted so that a double
pendulum of a lifted thigh and a passive shank-foot reaches a
step length of 0.7 m within 300 ms (typical values for normal
walking). Since the ankle push-off from a stand still does not
suffice, the driven pendulum physics require that the thigh
decelerates eventually allowing the inertia of the shank-foot
to passively rotate it around the thigh and bring the leg into
extension, which guided setting the reflex gain and offset of the
L- from HAM to HFL. The reflex gains of the F+ of GLU and
, which resulted in a gentle
HAM were initially set to
leg lowering and retraction of the double pendulum toward the
end of the step with 0.7 m length. Finally, the manual reflex
tuning that followed also delivered the necessary values for
the remaining reflexes including the suppressions of TA via Ffrom SOL and of VAS via knee angle feedback (Section II-A),
and the trunk bias of HFL’s L+ in swing (Section II-C). The
final value for each reflex parameter is shown in Table I along
with its sensitivity (a change during steady-state locomotion
beyond the min/max limits leads to a fall).
The equations below implement the reflex control computing
. All stimulations are limited
the muscle stimulations
.
from 0.01 to 1 before they produce muscle activations
The time delays of 20, 10, and 5 ms in the equations represent
long, medium and short neural signal delays. They were not
tuned but estimated from the time gaps between M-wave and
H-wave of H-reflex experiments (for details see [29]).
Stance Reflexes (
ms,
ms,
ms, DSup is 1 if leg is trailing leg in
and
refers to only posidouble support, otherwise 0,
;
tive/negative values):
;
;
;
;
;
.
Fig. 6. Muscle-tendon model. An active, contractile element (CE) together
with a series elasticity (SE) form the MTU in normal operation. If the CE
stretches beyond its optimum length (` > ` ), a parallel elasticity (PE)
engages. Conversely, a buffer elasticity (BE) prevents the active CE from
` <`
).
collapsing if the SE is slack (`
0
Swing reflexes (
previous take off):
:
constant
;
;
value
;
taken
at
;
;
;
.
APPENDIX II
MUSCLE TENDON UNITS
All 14 muscle-tendon units (MTUs) have a common model
structure (Fig. 6). An MTU’s force
is computed from resolving the inner degree of freedom
. With
,
is equal to
with
, where is the muscle activation,
,
and
are the
the maximum isometric force,
force-length and force-velocity relationships of the contractile
,
, and
are the forces of the series
element (CE), and
(SE), parallel (PE), and buffer elasticity (BE). Details on how
are given in [29]; for completeness,
we model , , , and
we here report the parameters required to compute these functions including the excitation-contraction coupling constant
of A; the width
and the residual force
of ; the eccentric force enhancement
factor
and the shape factor
of ; and the reference strain
of
.
, where
is the BE rest length and
is a reference compression.
with the PE reference strain
.
allows
with
to rewrite
, which can robustly be
integrated with coarse time steps, because it cannot run into
. Note that PE and BE engage
negative results
outside the normal range of operation of the MTU and play
minor roles for its dynamics in locomotion. The MTUs share
the same parameters except for four main ones that distinguish
individual muscle physiology (Table II).
GEYER AND HERR: A MUSCLE-REFLEX MODEL THAT ENCODES PRINCIPLES OF LEGGED MECHANICS
TABLE II
INDIVIDUAL MTU PARAMETERS. VALUES ARE ESTIMATED FROM [64]
ASSUMING A FORCE OF 25 N PER cm CROSS SECTIONAL AREA
(F
) MAXIMUM SPEEDS OF 6` s
AND 12`
s FOR SLOW
AND MEDIUM-FAST TWITCH MUSCLES (v
), AND `
AND
TO REFLECT MUSCLE FIBER AND TENDON LENGTHS
`
TABLE III
MTU ATTACHMENT PARAMETERS (VALUES MOTIVATED FROM [65]–[68]
OR ANATOMICAL ESTIMATES)
271
. This nonlinear spring-damper model is motivated from the literature [41], [69], but interprets contacts with
and maxtwo basic material properties: ground stiffness
imum relaxation speed
. Here,
or 0 describes
a perfectly elastic or inelastic impact. Note that we use the
same model for the joint soft limits with
and
. A CP’s horizontal GRF is modeled
as either sliding force
or stiction
, where
force
is the CP’s horizontal velocity,
, the sliding friction
, the horizontal contact stiffness,
coefficient,
, the shift from the point
at which stiction
, its velocity again normalized to
.A
engaged, and
and returns to sliding
CP engages in stiction if
with a stiction coefficient
.
if
ACKNOWLEDGMENT
TABLE IV
SEGMENT PARAMETERS (VALUES APPROXIMATED FROM [41])
The authors would like to thank A. Prochazka and the anonymous reviewers for helpful comments and suggestions, which
led to substantial improvements of the paper.
REFERENCES
APPENDIX III
MTU ATTACHMENTS AND SEGMENT PROPERTIES
The MTUs connect to the skeleton by spanning one or two
to joint
joints (Table III). The transfer from MTU force
is modeled as
, where the lever
torque
equals for the hip and
for the ankle
gets maximal at
and knee. Here, is the joint angle and
. Changes in MTU length are modeled as
for the hip and as
for the ankle and knee, where
is the joint angle at
, and accounts for muscle pennawhich
tion angles and ensures that the MTU fiber length stays within
physiological limits throughout the joint work space.
The model’s segments are rigid bodies specified by their mass
, inertia
, and length , and the positions
of the
of the proximal joint measured
local center of mass and
from the distal end (Table IV). The segments are connected by
,
revolute joints with ranges of operation,
and
, outside of which soft limits engage
(see Appendix IV).
APPENDIX IV
GROUND CONTACTS AND JOINT LIMITS
The model’s foot segments have toe and heel contact points (CPs). A CP’s vertical GRF is modeled as
,
is the vertical contact stiffness,
where
, ground penetration, and
, its velocity normalized to
[1] T. G. Brown, “On the nature of the fundamental activity of the nervous
centres; together with an analysis of the conditioning of rhythmic activity in progression, and a theory of the evolution of function in the
nervous system,” J. Physiol., vol. 48, no. 1, pp. 18–46, 1914.
[2] G. Orlovsky, T. Deliagina, and S. Grillner, Neuronal Control of Locomotion: From Mollusc to Man. New York: Oxford Univ. Press, 1999.
[3] I. A. Rybak, N. A. Shevtsova, M. Lafreniere-Roula, and D. A. McCrea,
“Modelling spinal circuitry involved in locomotor pattern generation:
Insights from deletions during fictive locomotion,” J. Physiol., vol. 577,
pt. 2, pp. 617–639, 2006.
[4] V. Dietz, “Spinal cord pattern generators for locomotion,” Clin. Neurophysiol., vol. 114, no. 8, pp. 1379–1389, 2003.
[5] K. Minassian, I. Persy, F. Rattay, M. M. Pinter, H. Kern, and M. R.
Dimitrijevic, “Human lumbar cord circuitries can be activated by extrinsic tonic input to generate locomotor-like activity,” Hum. Movement
Sci., vol. 26, no. 2, pp. 275–295, 2007.
[6] S. Grillner and P. Zangger, “On the central generation of locomotion in
the low spinal cat,” Exp. Brain Res., vol. 34, no. 2, pp. 241–261, 1979.
[7] K. Pearson, O. Ekeberg, and A. Buschges, “Assessing sensory function
in locomotor systems using neuro-mechanical simulations,” Trends
Neurosci., vol. 29, no. 11, pp. 625–631, 2006.
[8] B. W. Verdaasdonk, H. F. J. M. Koopman, and F. C. T. Van der Helm,
“Resonance tuning in a neuro-musculo-skeletal model of the forearm,”
Biol. Cybern., vol. 96, no. 2, pp. 165–180, 2007.
[9] O. Ekeberg and S. Grillner, “Simulations of neuromuscular control in
lamprey swimming,” Philos. Trans. R. Soc. Lond. B Biol. Sci., vol. 354,
no. 1385, pp. 895–902, 1999.
[10] A. Ijspeert, A. Crespi, D. Ryczko, and J.-M. Cabelguen, “From
swimming to walking with a salamander robot driven by a spinal cord
model,” Science, vol. 315, no. 5817, pp. 1416–1420, 2007.
[11] D. G. Ivashko, B. I. Prilutski, S. N. Markin, J. K. Chapin, and I.
A. Rybak, “Modeling the spinal cord neural circuitry controlling cat
hindlimb movement during locomotion,” Neurocomputing, vol. 52–54,
pp. 621–629, 2003.
[12] A. Prochazka and S. Yakovenko, “The neuromechanical tuning hypothesis,” Prog. Brain Res., vol. 165, pp. 255–265, 2007.
[13] C. Maufroy, H. Kimura, and K. Takase, “Towards a general neural controller for quadrupedal locomotion,” Neural Netw., vol. 21, no. 4, pp.
667–681, 2008.
[14] N. Ogihara and N. Yamazaki, “Generation of human bipedal locomotion by a bio-mimetic neuro-musculo-skeletal model,” Biol. Cybern.,
vol. 84, no. 1, pp. 1–11, 2001.
272
IEEE TRANSACTIONS ON NEURAL SYSTEMS AND REHABILITATION ENGINEERING, VOL. 18, NO. 3, JUNE 2010
[15] C. Paul, M. Bellotti, S. Jezernik, and A. Curt, “Development of a human
neuro-musculo-skeletal model for investigation of spinal cord injury,”
Biol. Cybern., vol. 93, no. 3, pp. 153–170, 2005.
[16] S. Mochon and T. McMahon, “Ballistic walking,” J. Biomech., vol. 13,
no. 1, pp. 49–57, 1980.
[17] T. McGeer, Principles of Walking and Running, ser. Advances in Comparative and Environmental Physiology. Berlin, Germany: SpringerVerlag, 1992, vol. 11, ch. 4.
[18] R. Blickhan, “The spring-mass model for running and hopping,” J.
Biomech., vol. 22, pp. 1217–1227, 1989.
[19] T. McMahon and G. Cheng, “The mechanism of running: How does
stiffness couple with speed?,” J. Biomech., vol. 23, pp. 65–78, 1990.
[20] T. McGeer, “Passive dynamic walking,” Int. J. Rob. Res., vol. 9, no. 2,
pp. 62–82, 1990.
[21] A. Seyfarth, H. Geyer, M. Günther, and R. Blickhan, “A movement
criterion for running,” J. Biomech., vol. 35, pp. 649–655, 2002.
[22] R. Ghigliazza, R. Altendorfer, P. Holmes, and D. Koditschek, “A
simply stabilized running model,” SIAM J. Appl. Dynamical Syst., vol.
2, no. 2, pp. 187–218, 2003.
[23] A. Seyfarth, H. Geyer, and H. M. Herr, “Swing-leg retraction: A
simple control model for stable running,” J. Exp. Biol., vol. 206, pp.
2547–2555, 2003.
[24] M. Raibert, Legged Robots That Balance. Cambridge, MA: MIT
Press, 1986.
[25] U. Saranli, M. Buehler, and D. Koditschek, “Rhex: A simple and highly
mobile hexapod robot,” Int. J. Rob. Res., vol. 20, no. 7, pp. 616–631,
2001.
[26] S. Collins, A. Ruina, R. Tedrake, and M. Wisse, “Efficient bipedal
robots based on passive-dynamic walkers,” Science, vol. 307, no. 5712,
pp. 1082–1085, 2005.
[27] H. Geyer, A. Seyfarth, and R. Blickhan, “Compliant leg behaviour explains the basic dynamics of walking and running,” Proc. R. Soc. Lond.
B, vol. 273, pp. 2861–2867, 2006.
[28] A. Prochazka, D. Gillard, and D. Bennett, “Positive force feedback control of muscles,” J. Neurophys., vol. 77, pp. 3226–3236, 1997.
[29] H. Geyer, A. Seyfarth, and R. Blickhan, “Positive force feedback in
bouncing gaits?,” Proc. R. Soc. Lond. B, vol. 270, pp. 2173–2183, 2003.
[30] A. Seyfarth, M. Günther, and R. Blickhan, “Stable operation of an
elastic three-segmented leg,” Biol. Cybern., vol. 84, pp. 365–382, 2001.
[31] M. Günther, V. Keppler, A. Seyfarth, and R. Blickhan, “Human leg
design: Optimal axial alignment under constraints,” J. Math. Biol., vol.
48, pp. 623–646, 2004.
[32] H. Herr and T. McMahon, “A trotting horse model,” Int. J. Robot. Res.,
vol. 19, pp. 566–581, 2000.
[33] H. Herr and T. McMahon, “A galloping horse model,” Int. J. Robot.
Res., vol. 20, pp. 26–37, 2001.
[34] H. M. Herr, G. T. Huang, and T. A. McMahon, “A model of scale
effects in mammalian quadrupedal running,” J. Exp Biol., vol. 205, pt.
7, pp. 959–967, Apr. 2002.
[35] K. Pearson and D. Collins, “Reversal of the influence of group Ib afferents from plantaris on activity in medial gastrocnemius muscle during
locomotor activity,” J. Neurophys., vol. 70, pp. 1009–1017, 1993.
[36] M. J. Grey, J. B. Nielsen, N. Mazzaro, and T. Sinkjaer, “Positive force
feedback in human walking,” J. Physiol., vol. 581, no. 1, pp. 99–105,
2007.
[37] L. O. Christensen, J. B. Andersen, T. Sinkjaer, and J. Nielsen, “Transcranial magnetic stimulation and stretch reflexes in the tibialis anterior muscle during human walking,” J. Physiol., vol. 531, pt. 2, pp.
545–557, 2001.
[38] N. Petersen, H. Morita, and J. Nielsen, “Modulation of reciprocal inhibition between ankle extensors and flexors during walking in man,” J.
Physiol., vol. 520, pt. 2, pp. 605–619, 1999.
[39] T. Mergner, C. Maurer, and R. J. Peterka, “A multisensory posture control model of human upright stance,” Prog. Brain Res, vol. 142, pp.
189–201, 2003.
[40] S. M. O’Connor and A. D. Kuo, “Direction dependent control of
balance during walking and standing,” J. Neurophysiol., vol. 102, pp.
1411–1419, 2009.
[41] M. Günther and H. Ruder, “Synthesis of two-dimensional human
walking: A test of the -model,” Biol. Cybern., vol. 89, pp. 89–106,
2003.
[42] V. T. Inman, H. J. Ralston, F. Todd, and J. C. Lieberman, Human
Walking. Baltimore, MD: Williams & Wilkins, 1981.
[43] V. Dietz and S. J. Harkema, “Locomotor activity in spinal cord-injured
persons,” J. Appl. Physiol., vol. 96, no. 5, pp. 1954–1960, 2004.
[44] M. Y. C. Pang and J. F. Yang, “Sensory gating for the initiation of
the swing phase in different directions of human infant stepping,” J.
Neurosci., vol. 22, no. 13, pp. 5734–5740, 2002.
[45] C. Schneider, M. C. Do, and B. Bussel, “Increase of the contraction of
the stance soleus muscle in human does not delay the swing phase in
step elicited by forward fall,” C R Acad. Sci. III, vol. 320, no. 9, pp.
709–714, 1997.
[46] S. N. Zill, B. R. Keller, and E. R. Duke, “Sensory signals of unloading
in one leg follow stance onset in another leg: Transfer of load and emergent coordination in cockroach walking,” J. Neurophysiol., vol. 101,
no. 5, pp. 2297–2304, 2009.
[47] T. Lam and K. G. Pearson, “Proprioceptive modulation of hip flexor
activity during the swing phase of locomotion in decerebrate cats,” J.
Neurophysiol., vol. 86, no. 3, pp. 1321–1332, Sep. 2001.
[48] J. J. Eng, D. A. Winter, and A. E. Patla, “Strategies for recovery from
a trip in early and late swing during human walking,” Exp. Brain Res,
vol. 102, no. 2, pp. 339–349, 1994.
[49] H. W. Van de Crommert, M. Faist, W. Berger, and J. Duysens, “Biceps
femoris tendon jerk reflexes are enhanced at the end of the swing phase
in humans,” Brain Res, vol. 734, no. 1–2, pp. 341–344, 1996.
[50] J. Perry, Gait Analysis: Normal and Pathological Function. Thorofare, NJ: SLACK Inc., 1992.
[51] L. Li and G. E. Caldwell, “Coefficient of cross correlation and the time
domain correspondence,” J. Electromyogr. Kinesiol., vol. 9, no. 6, pp.
385–389, 1999.
[52] T. A. L. Wren, K. P. Do, S. A. Rethlefsen, and B. Healy, “Cross-correlation as a method for comparing dynamic electromyography signals
during gait,” J. Biomech., vol. 39, no. 14, pp. 2714–2718, 2006.
[53] J. U. Baumann, H. Ruetsch, and K. Schürmann, “Distal hamstring
lengthening in cerebral palsy. An evaluation by gait analysis,” Int.
Orthop., vol. 3, no. 4, pp. 305–309, 1980.
[54] A. J. Ijspeert, “Central pattern generators for locomotion control in animals and robots: A review,” Neural Netw., vol. 21, no. 4, pp. 642–653,
2008.
[55] L. S. Illis, “Is there a central pattern generator in man?,” Paraplegia,
vol. 33, no. 5, pp. 239–240, 1995.
[56] A. Lundberg, “Reflex control of stepping,” The Nansen Memorial Lecture V. Oslo, Norway, Universitetsforlaget, pp. 5–42, 1969.
[57] G. Taga, “A model of the neuro-musculo-skeletal system for human
locomotion. I. Emergence of basic gait,” Biol. Cybern., vol. 73, no. 2,
pp. 97–111, 1995.
[58] K. G. Pearson, “Generating the walking gait: Role of sensory feedback,” Prog. Brain Res., vol. 143, pp. 123–129, 2004.
[59] H. Hultborn, “Spinal reflexes, mechanisms and concepts: From Eccles to Lundberg and beyond,” Prog. Neurobiol., vol. 78, no. 3–5, pp.
215–232, 2006.
[60] A. Prochazka, V. Gritsenko, and S. Yakovenko, “Sensory control of locomotion: Reflexes versus higher-level control,” Adv. Exp. Med. Biol.,
vol. 508, pp. 357–367, 2002.
[61] J. M. Donelan, D. A. McVea, and K. G. Pearson, “Force regulation of
ankle extensor muscle activity in freely walking cats,” J. Neurophysiol.,
vol. 101, no. 1, pp. 360–371, 2009.
[62] T. Sinkjaer, J. B. Andersen, M. Ladouceur, L. O. Christensen, and J.
B. Nielsen, “Major role for sensory feedback in soleus EMG activity
in the stance phase of walking in man,” J. Physiol., vol. 523, pt. 3, pp.
817–827, 2000.
[63] M. A. Daley, G. Felix, and A. A. Biewener, “Running stability is enhanced by a proximo-distal gradient in joint neuromechanical control,”
J. Exp. Biol., vol. 210, pt. 3, pp. 383–394, 2007.
[64] G. T. Yamaguchi, A. G.-U. Sawa, D. W. Moran, M. J. Fessler, and J.
M. Winters, “A survey of human musculotendon actuator parameters,”
in in Multiple Muscle Systems: Biomechanics and Movement Organization, J. Winters and S.-Y. Woo, Eds. New York: Springer-Verlag,
1990, pp. 717–778.
[65] T. Muraoka, Y. Kawakami, M. Tachi, and T. Fukunaga, “Muscle fiber
and tendon length changes in the human vastus lateralis during slow
pedaling,” J. Appl. Physiol., vol. 91, pp. 2035–2040, 2001.
[66] C. Maganaris, “Force-length characteristics of in vivo human skeletal
muscle,” Acta Physiol. Scand., vol. 172, pp. 279–285, 2001.
[67] C. Maganaris, “Force-length characteristics of the in vivo human gastrocnemius muscle,” Clin. Anat., vol. 16, pp. 215–223, 2003.
GEYER AND HERR: A MUSCLE-REFLEX MODEL THAT ENCODES PRINCIPLES OF LEGGED MECHANICS
[68] T. Oda, H. Kanehisa, K. Chino, T. Kurihara, T. Nagayoshi, E. Kato,
T. Fukunaga, and Y. Kawakami, “In vivo lenth-force relationships
on muscle fiver and muscle tendon complex in the tibialis anterior
muscle,” Int. J. Sport Health Sci., vol. 3, pp. 245–252, 2005.
[69] S. Scott and D. Winter, “Biomechanical model of the human foot: Kinematics and kinetics during the stance phase of walking,” J. Biomech.,
vol. 26, no. 9, pp. 1091–1104, 1993.
Hartmut Geyer received the Dipl. degree in physics
and the Ph.D. degree in biomechanics from the
Friedrich-Schiller-University of Jena, Germany, in
2001 and 2005, respectively.
He is Assistant Professor at the Robotics Institute
of Carnegie Mellon University, Pittsburgh, PA. His
research focuses on the principles of legged dynamics
and control, their relation to human motor control,
and resulting applications in rehabilitation robotics.
273
Hugh Herr (M’08) received the B.A. degree in
physics from Millersville University, Millersville,
PA, in 1990, the M.S. degree in Mechanical Engineering from the Massachusetts of Technology
(MIT), Cambridge, and the Ph.D. in biophysics from
Harvard University, Cambridge, MA, in 1998.
He is Associate Professor within MIT’s Program
of Media Arts and Sciences, and The Harvard-MIT
Division of Health Sciences and Technology. His
primary research objective is to apply principles of
biomechanics and neural control to guide the designs
of prostheses, orthoses, and exoskeletons. He is the author of over 60 technical
publications in biomechanics and wearable robotics.
Dr. Herr is the recipient of the 2007 Heinz Award for Technology, the
Economy and Employment.